Frequency Analysis

As shown in this tour, the notion of instantaneous frequency should be handled with caution. This is why we recall the primary definition of a frequency.

Note: this presentation is proper to the site and does not come from the book.

Period and Frequency

What is a frequency?

A frequency is the inverse of a period.

A one dimensional signal is T>0 periodic if it is unchanged by a translation of T. Hence its support has an infinite length. Nonetheless, the signal is entirely determined by its values over an interval of length T.

In the representation over an interval of length T, the regularity of the signal implies the equality of the values of the functions and of its derivatives at the left and right ends of the interval.

A musical example

A purely synthetic music note can be represented by a sinusoidal wave. An instrumental note that is held is a more complex signal. A music or a speech recording is even more complex; in particular, the frequencies may vary with time.

As a first approximation, a music piece may be modelized as a sequence of signals defined over intervals whose length is determined by the tempo. Every elementary signal can be periodized.

Fourier Analysis

Classically, the analysis of a signal as a Fourier series or a Fourier integral provides a representation of its frequency contents.

Fourier series